Sum of disjoint product approach for reliability evaluation of stochastic flow networks (original) (raw)
Related papers
Efficient Estimation of Stochastic Flow Network Reliability
IEEE Transactions on Reliability, 2019
The Creation Process is an algorithm that transforms a static network model into a dynamic one. It is the basis of different variance reduction methods designed to make efficient reliability estimations on highly reliable networks in which links can only assume two possible values, operational or failed. In this article the Creation Process is extended to let it operate on network models in which links can assume more than two values. The proposed algorithm, called here Multi-Level Creation Process, is the basis of a method, also introduced here, to make efficient reliability estimations of highly reliable stochastic flow networks. The method proposed, which consists in an application of Splitting over the Multi-Level Creation Process, is empirically shown to be accurate, efficient, and robust.
Reliability evaluation of a stochastic-flow network (SFN) has been extensively studied in the past decades and various algorithms have been proposed. A number of graph based algorithms are in terms of minimal paths (MPs). Most MP-based algorithms have been proposed for the case of single-source single-sink SFN and a few algorithms have considered the multi-source multi-sink case. However, there are many practical networks such as communication and telecommunication networks comprised of several sources and sinks. Here, we consider a multi-source multi-sink SFN and propose an algorithm to find all the lower boundary points in terms of the MPs. The proposed algorithm is shown to be correct. Moreover, the algorithm is shown to be more efficient than some existing ones. After World War I, reliability was measured as the number of accidents per hour of flight time for one-, two-, and four-engine airplanes [1]. Afterwards, network reliability theory has extensively been applied to a variety of real-world systems such as power transmission and distribution [2], mobile ad hoc wireless [3], transportation [4], and manufacturing [5]. Moreover, in problems such as maximizing system reliability [2, 5] or optimal design of a network subject to reliability constraint [6], there has been an increasingly significant need for efficiently computing or estimating the system reliability. Thus, the system reliability problem turns to be an important challenging problem for system engineers. Evaluating the system reliability is an NP-hard problem [7], and thus the problem continues to be interesting to investigate. In a single-source single-sink stochastic-flow network (SS-SFN), system reliability is usually considered as the probability of transmitting a given amount of flow (or data) from a source node to a sink node [7−14]. A number of graph-based algorithms have been proposed in terms of minimal cuts (MCs) [7−11] or minimal paths (MPs) [2, 3, 12, 13] to evaluate the system reliability of an SS-SFN. Forghani-elahabad and Mahdavi-Amiri [8] considered an SS-SFN with budget constraint and proposed an effective algorithm to evaluate the reliability of the network. In [9], investigating several existing algorithms, their flaws were illustrated and modifying the flaws, an improved algorithm was proposed to compute the exact system reliability. In addition, Forghani-elahabad and Mahdavi-Amiri [10] presented a new data structure along with some new results to propose an improved algorithm. The authors showed the algorithm to be more efficient than the other existing algorithms theoretically and practically. In [11], an improved algorithm was proposed to determine all the upper boundary points in order to compute the system reliability. Later, the authors in [12] considered the case of sending all the flow through two separate minimal paths (SMPs), and proposed an algorithm to find the two most reliable SMPs transmitting a given amount of flow within the given time and budget constraints. They also extended the proposed algorithm to the case of q SMPs in [13]. In a multi-source multi-sink stochastic-flow network (MM-SFN), there is also a demand for flow to be transmitted from each source to its corresponding sink, and hence the system reliability in an MM-SFN is the probability of transmitting all the required demands from the sources to their corresponding sinks simultaneously. Lin and Yuan [15] considered an MM-SFN and proposed an algorithm to evaluate the system reliability in terms of minimal paths. Considering a real-world network, namely Taiwan Advanced Research and Education Network, Lin and Yen [16] proposed an algorithm
A survey of efficient reliability computation using disjoint products approach
Networks, 1995
Several algorithms have been developed to solve the reliability problem for nonseries-parallel networks using the sum of disjoint products (SDP) approach. This paper provides a general framework for most of these techniques. It reviews methods that help improve computer time and memory requirements in reliability computation. These parameters are generally used to compare SDP algorithms. We also overview three multiple variable inversion algorithms that result in sum of disjoint products expressions with fewer terms than that of algorithms that use only a single-variable inversion. One common network is solved for twoterminal network reliability using each of these algorithms. Finally, we have provided a comparison among these techniques. 0
On reliability evaluation of a capacitated-flow network in terms of minimal pathsets
Networks, 1995
Many real-world systems such as electric power transmission and distribution systems, transportation systems, and manufacturing systems can be regarded as flow networks whose arcs have independent, finite, and multivalued random capacities. Such a flow network is indeed a multistate system with multistate components and so its reliability for the system demand d , i.e., the probability that the maximal flow is no less than d , can be computed in terms of minimal path vectors to level d (named d-MPs here). The main objective of this paper was to present a simple algorithm to generate all d-MPs of such a system for each system capacity level d in terms of minimal pathsets. Analysis of our algorithm and comparison to Xue's algorithm shows that our method has the following advantages: (1) the family of d-MP candidates that it generates is smaller in size and so d-MPs can be generated more efficiently, (2) it is expressed more intuitively and so easier to understand, and (3) whenever applied in a seriesparallel case, both algorithms are essentially the same, but in a non series-parallel case, Xue's algorithm needs the extra work to transform the system into a series-parallel in advance. Two examples are illustrated to show how all d-MPs are generated by our algorithm and then the reliability of one example is computed. 0 7995 John Wiley & Sons, Inc.
Efficient Algorithms for Reliability Evaluation of General Networks
Frontiers in Science and Engineering, 2016
Several production systems either for goods or services can be modeled by a network where nodes are production centers, warehouses, distributions and others, and arcs represent the relationship between nodes. Nodes and arcs are often subjected to random failures that may result from several causes. These networks include one or more sources and one or more destinations. Given the stochastic nature of the failure, the reliability and the robustness of the network become an important criteria for safety, economical and environment reasons. Several methods based on graph theory and stochastic processes are proposed in the literature. The concepts of minimal paths set (MPS) and minimal cuts set (MCS) as well as decomposition techniques based on Bayes' theorem have been widely used. The performance of these methods is greatly affected by network size (number of nodes and arcs) and its density. Generally, except for special structure of some networks (e.g series, parallel, standby, etc.) there is no mathematical expression based on the reliability of its nodes and its arcs that has been proved compact for representing the expression of the reliability function of any network. This paper attempts to provide solutions to this problem by proposing and testing a unified approach based on MPS/MCS and Binary Decision Diagrams (BDD). This approach is illustrated by several simple examples. A tool has been developed to handle complex networks such as telecommunication networks and other network's tests published in the literature.
Reliability evaluation of a limited-flow network in terms of minimal cutsets
IEEE Transactions on Reliability, 1993
& Conclusions-Many systems can be regarded as flow networks whose arcs have discrete and multi-valued random capacities. The probability of the maximum flow at each various level and the reliability of such a flow network can be calculated in terms of K-lattices which are generated from each subset of the family of all MCs (minimal cutsets). However the size of such a family 2m-1 (m = number of MCs) grows exponentially with m. Such a flow network can be considered as a multistate system with multistate components so that its reliability can be evaluated in terms of upper boundary points of each level d (named d-MCs here). This article presents an algosthm to generate all d-MCs from each MC for each system capacity level d. After analyzing and comparing it with the algorithm by Xue, it ensures that our method generates a family of d-MC candidates which contains all d-MCs more efficiently if both start from MCs. Examples show how all d-MCs are generated; the reliability of one example is computed. 'Editors' note: To allow clear, easy reference, we have named the Xue algorithm [18] "XGMS" (Xue general multistate sytem), and the algorithm proposed "JLY" (Jane, Lin, Yuan).
An MP-based approximation algorithm on reliability evaluation of multistate flow networks
Reliability Engineering & System Safety, 2019
In recent decades, multistate two-terminal reliability problem has attracted several researchers, and accordingly many exact and approximation approaches have been proposed in the literature in terms of minimal cuts (MCs) or minimal paths (MPs) to address this problem. Here, an MPbased approximation approach is developed based on exact algorithms. With all the MPs at hand, the approach rearranges the MPs ascendingly with respect to their lengths and then sets the flow on some MPs to be zero which turns to reduce the computing cost in solving the problem. We provide the complexity results, and by employing some benchmarks and one thousand randomly generated networks illustrate that not only in many cases the proposed approach determines very good approximate solutions much faster than the exact algorithms, but also in many other cases it even determines exact solutions significantly faster than the available exact algorithms in the literature. Moreover, the Dolan-Moré performance profile affirms the efficiency of our proposed algorithm. Finally, we state how to compute the system reliability by using the d-MP s, and show that from a very good approximation set of d-MP s, the system reliability is approximated with a very good accuracy.
International Journal of System Assurance Engineering and Management, 2019
Computer and communication applications such as big data analytics services, social networks, enterprise business and mobile applications often require to simultaneously pass data of different sizes between multiple source and multiple destinations in the network. In practice, the nodes in the communication networks are often connected through multiple intermediate links of different types with varying failure probabilities. Failure of intermediate links can adversely affect data transmission between source and destination nodes; thus, they can impact the quality of service. Therefore, reliability evaluation of such networks is of subtle importance in today's service dependent world. This article presents an efficient method for reliability evaluation of stochastic flow networks that can pass various demands simultaneously from multiple source nodes to multiple destination nodes. The proposed method has three steps: First step obtains the combined minimal cut sets between the given set of source and destination nodes. Second step generates the set of simultaneous upper boundary flows for the varying demands using the combined minimal cut sets. Third step calculates the network unreliability by applying the Sum of Disjoint Product method on the upper boundary flows. The reliability of the network, i.e., the probability that the network can simultaneously pass the set of demands, is calculated as 1-Unreliability. The MATLAB simulation of the proposed method on bench mark networks show that the proposed approach takes less computational time than the existing method. Keywords Stochastic flow networks Á Multi-source multidestination Á Minimal cut sets Á Combined cut sets Á Upper boundary flow vectors Á Sum of disjoint products
Reliability Evaluation of Multi-State Flow Networks Via Map Methods
Journal of Engineering Research and Reports, 2020
This paper examines two simple (albeit useful) methods used to evaluate the reliability of two-terminal multistate flow networks. These two methods involve two Karnaugh map versions, namely the Variable-Entered Karnaugh Map (VEKM) and the Multi-Valued Karnaugh Map (MVKM). These two versions are crucial in providing not only the visual insight necessary to write better future software but also adequate means of verifying such software. We assess these two versions of map methods versus the exhaustive search method, which guarantees conceptual clarity at the expense of lack of computational efficiency. Our target is the evaluation of the probability mass function (pmf) in a wide array of cases, in which we consider flow from a source node to a sink node in a capacitated network with a multistate capacity model for the links. Each network link has a varying capacity, which is assumed to exist in a mutually exclusive sense. The reliability of the system is wholly dependent on its ability to successfully transmit at least a certain required system flow from the source (transmitter) to the sink (receiver) station. The max-flow min-cut theorem is critical in obtaining all successful states. To demonstrate the proposed methods applicability, two demonstrative examples are given with ample details.
Reliability Evaluation of Multicommodity Limited-Flow Networks with Budget Constraints
—Many real-world systems such as manufacturing systems, transportation systems and logistics/distribution systems that play important roles in our modern society can be regarded as multicommodity flow networks whose arcs have independent, finite and multi-valued random capacities. Such a flow network is a multistate system with multistate components and its reliability for level (d;c), i.e., the probability that k different types of commodity can be transmitted from the source node to the sink node such that the demand level) ,..., , (2 1 k d d d d is satisfied and the total transmission cost is less than or equal to c, can be evaluated in terms of minimal path vectors to level (d;c) (named (d;c)-MPs here). The main objective of this paper is to present an intuitive algorithm to generate all (d;c)-MPs of such a flow network for each level (d;c) in terms of minimal pathsets. Two examples are given to illustrate how all (d;c)-MPs are generated by our algorithm and then the reliability of one example is computed.